Chiral Anomaly and the BaBar Measurements

of the γγ∗→ π

0 Transition Form Factor

T. N. PHAM

Centre de Physique Theorique,

Centre National de la Recherche Scientifique,

Ecole Polytechnique, 91128 Palaiseau Cedex, France

QCD@Work- International Workshop on QCD, Theory and Experiment,

Lecce, Italy, June 18-21, 2012.

Based on works with X. Y. Pham:

Phys. Lett. B 247, 438 (1990) , Int. J. Mod. Phys. A26, 4125 (2011).

1

1 Introduction

• The γ∗γ → π0 transition form factor F (Q2) at large virtual photon

momentum transfer Q2 is one of the simplest quantities to compute in

QCD.

• Relatively easy to measure, as in the e+e− → e+e−π0 single-tagged

experiment at Babar.

• At Q2 = 0, F (Q2) is given by the two-photon π0 decay governed by the

Adler-Bell-Jackiw(ABJ) triangle chiral anomaly which gives correctly the

decay rate.

• At large Q2, short-distance operator expansion(OPE) [Frishman] or

perturbative QCD [Brodsky-Lepage,Kroll] predicts F (Q2) ∼ 2 fπ/Q2 (

fπ = 93MeV ).

2

• The γ∗γ → π0 transition form factor in pQCD(Brodsky-Lepage et al)

Fπγ(Q2) = 2√Nc(e

2u − e2d)

∫ 1

0

dx1dx2

∫

∞

0

d2k⊥16π2

ψ(xi, k⊥)

[

(q⊥x2 + k⊥) × ε⊥(q1 × ε⊥)(q⊥x2 + k⊥)2

+ (x1 ↔ x2)

]

(1)

• Wave function peaked at low k⊥

• At large Q2, k⊥ � q⊥, by neglecting k⊥, one has:

Fπγ(Q2) =2√Nc(e

2u − e2d)

Q2

∫ 1

0

dx1dx2

x1x2

∫ Q2

0

d2k⊥16π2

ψ(xi, k⊥) (2)

• The quark distribution amplitude(DA) φ(xi, Q) : amplitude for finding

the constituent quark with longitudinal momenta xi :

φ(xi, Q) = (lnQ2

Λ2)−γF /β

∫ Q2

0

d2k⊥16π2

ψ(xi, k⊥) (3)

3

• From evolution equation, Brodsky et al obtain solution for φ(xi, Q),

φ(xi, Q) = x1x2

∞∑

n=0

anC3/2n (ln

Q2

Λ2)−γn (4)

• Q2 → ∞, only a0 survives, one has the asymptoptic limit:

Fπγ(Q2) =2(fπ)

Q2(5)

p p’(tag)

π0q1

q2

p

p’

q1

q2

π0

Figure 1: Diagrams for e+e−

→ e+e−π0 and e+e−

→ π0γ

4

• The earlier CLEO data for F (Q2) up to Q2 = 8GeV2 somewhat below

the perturbative QCD(pQCD) prediction, though, with a possible rise

for Q2 F (Q2) above 2.5GeV2.

• The BaBar Collaboration has produced measurements for Q2 from 4 to

34 GeV2 [BaBar] which show spectacular deviation from the

perturbative QCD prediction as seen from the data for Q2 F (Q2).

• Q2 F (Q2) of BaBar rises steadily with Q2 in contrast with the rather

flat behavior predicted by pQCD and is more than 50% above the QCD

prediction at 34GeV2 [Brodsky-Lepage]

• The new Belle results, though somewhat below the BaBar data,

indicates some rise with Q2 for Q2 F (Q2) at large Q2.

5

CELLOCLEOBABAR

Q2 (GeV2)

Q2 |F

(Q2 )|

(GeV

)

0

0.1

0.2

0.3

0 10 20 30 40

Figure 2: BaBar data for the transition form factor multiplied by Q2 taken from

BaBar published PRD 80,052002 (2009)

6

CELLOCLEOBABAR

CZ

ASY

BMS

Q2 (GeV2)

Q2 |F

(Q2 )|

(GeV

)

0

0.1

0.2

0.3

0 10 20 30 40

Figure 3: BaBar data for Q2 F (Q2) compared with current theoretical predictions

taken from BaBar PRD paper

7

Figure 4: The Belle results for Q2 F (Q2) compared with the BaBar measurements

taken from Belle paper

8

• Recent calculations using the light-cone sum rules method at

next-to-leading order with various forms for the pion distribution

amplitude, seem to obtain values for the transition form factor higher

than the asymptotic limit, but with very different Q2 behavior than the

BaBar data for Q2 < 15GeV2 and are below the BaBar data for Q2 in

the range from 20 to 40 GeV2

• As pointed out by BaBar, existing calculations with

Chernyak-Zhinitsky(CZ) DA , asymptotic DA(ASY),

Bakulev-Mikhailov-Stefanis DA (BMS) for pion, produce only a flat Q2

dependence for Q2 > 10GeV2

• The rise with Q2 for Q2 F (Q2) indicates the presence of hard

component in the pion distribution amplitude(DA).

• More recent papers with broad DA distribution, ( Huang and Wu

(2009), Radyushkin (2009)) and the latest model with flat DA(Agnev,

Braun, Offen and Porkett (2011)) could produce a log(Q2) rise of

Q2 F (Q2) for Q2 > 15GeV2, though still somewhat below the BaBar

data, but could explain the new Belle results at large Q2 above 30GeV2

9

Q2 Fγγ∗π0(Q2) (MeV )

150

200

250

300

20 30 40

Q2 (GeV 2)

10

Figure 5: prediction of Radyushkin

10

0 10 20 30 400

0.1

0.2

0.3

0.4

Q2 (GeV2)

Q2 F

πγ(Q

2 ) (G

eV)

BaBar dataCELLO dataCLEO dataBelle data

Figure 6: prediction of Wu et al, 2012

• The flat DA distribution model with Sudakov suppression and kT

factorization, (the modified perturbation approach(MPA) of Li and

Mishima(2009), Kroll(2011)) seems to explain the BaBar data large Q2

11

but with extra Q2 dependence in the Sudakov suppression factor in the

quantity c(Q).

]2 [GeV2Q0 10 20 30 40 50

) [G

eV]

2(Q γπ

F2

Q

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

c=0.3

)2 c(Q

Figure 7: prediction of Li and Mishima with flat DA in kT factorization

12

• Similarly, in the very recent paper, Agnev et al would also need a flat

DA or a large contribution from the large invariant mass in the dispersion

representation of the transition form factor to explain the BaBar data.

óóóóóóóó

óóóóó

èèèèè è èè è è

èè è è

è

è

è

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Figure 8: Q2F (Q2) obtained by Agnev et al for a flat pion DA with reduced second

Gegenbauer coefficient aflat2 = 0.130 (model 1)

13

èèèèè è èè è è

èè è è

è

è

è

óóóóóóóó

óóóóó

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Figure 9: Q2F (Q2) from LCSR obtained by Agnev et al for 3 models with flat pion

DA .with aflat2 = 0.130. The curves showing the contribution from large(hard)(middle

curve) and small(soft)(lower curve) invariant mass in the dispersion representation

and the total hard + soft contribution(the upper curve)

14

• As pointed out by Dorokhov, a flat DA for the pion corresponds to a

point-like coupling of pion to quark. This produces a log(Q2) increase

with large Q2 in Q2F (Q2) which is however still somewhat below the

BaBar data for Q2 > 15GeV2.

• In a previous work, we have shown(the PLB 1990 paper) that the

transition form factor γ∗γ → π0 with one virtual photon with space-like

or time-like Q2 computed using PCAC and the Adler-Bell-Jackiw chiral

anomaly, for large Q2, behaves as (log(Q2)2/Q2 , faster than the simple

log(Q2) in recent pQCD calculations.

• This work: apply our previous result to to the γ∗γ → π0 transition

form factor.

15

2 Chiral anomaly effects for the

γ∗γ → π

0 transition form factor

• PCAC and Adler-Bell-Jackiw chiral anomaly

The success of the Goldberger-Treiman relation for the pion-nucleon

coupling constant obtained from the PCAC hypothesis shows that

SU(2) × SU(2) is a good symmetry for strong interactions.

• Pion is an almost Nambu-Goldstone boson generated by the

spontaneous breakdown of chiral SU(2) × SU(2). The success of the

chiral anomaly prediction for π0 → γγ decay is a confirmation of the

existence of Adler-Bell-Jackiw chiral anomaly in a theory with quarks

and gluons.

16

Some predictions Data

M2π/M2

p = 0 0.03

2 MgA/(fπgpnπ) + 1 = 0 0.06 ± 0.01

M2πa

12 = 0.16 0.17 ± 0.005

M2πa

32 = −0.078 −0.088 ± 0.004

λKe3= 0.021 ± 0.003 0.019 ± 0.004

Γ(π0→ γγ) = 7.87 eV 7.95 ± 0.55

Table 1: Chiral symmetry and PCAC predictions(Taken from C. H. Llewllyn Smith,

Proc. of the 1989 Scottish Universities Summer School Physics of the Early Uni-

verse)

• One can derive the Goldberger-Treiman(GT) relation by going to the

exact chiral symmetry limit: mu,d = 0, ∂µAµ = 0 and obtain

2mNgA(q2) + q2fP (q2) = 0 (6)

for the matrix element of the isovector axial vector current between

nucleon states.

17

• Since Mn 6= 0, fP (q2) must have a pole at q2 = 0. This corresponds to

a massless pion since it couples to the nucleon through the pion-nucleon

coupling constant:

fP (q2) = 2gπNfπ/(−q2) (7)

• Like the π0 → γγ decay, PCAC is modified by the Adler-Bell-Jackiw

triangle anomaly.

• Historically, Jacob and Wu apply the modifed PCAC to high energy

processes like Z0 → π0γ and W± → π±γ decays

• Suppression of the process Z0 → π0γ due to cancellation of the

anomaly by the axial current matrix element in the triangle graph.

• There remains a (log(Q2))2/Q2 term for the γ∗γ → π0 transition form

factor. Bando and Harada(1994) ; Hayakawa and Kinoshita(1998) also

obtained (log(Q2))2/Q2 for the γ∗γ → π0 transisiton transition form

factor from the chiral anomaly.

18

A(p)

q, q^2=-Q^2 q,q^2=-Q^2

k,k^2=0k,k^2=0

A(p) π0(p)

Figure 10: direct and pion pole terms in the triangle graph for γ∗γ → π0 transisiton

transition form factor

• This work: apply our result to the γ∗γ → π0 transition form factor.

• Modified PCAC by ABJ anomaly

∂µAµ = fπm

2πφ+ S

e2

16π2εαβγδF

αβF γδ (8)

with S = 1/2 in the SM.

• The transition form factor F (q, k) is defined as

Nµν(q, k) = e2F (q, k)Y µν , Y µν = εµναβqαkβ . (9)

19

• The matrix element < 0|Aµ|γ∗γ > is the sum of the direct and the

pion pole term.

• The pion term is then:

Nµν =1

fπ

(

pτ Rµντ (q, k) − S

e2

2π2Y µν

)

(10)

• The direct term is

pτ Rµντ (q, k) = e2S

(

2mP (q, k) +1

2π2

)

Y µν (11)

with

P (q, k) =m

2π2

∫ 1

0

dx

∫ 1−x

0

dy

D

D = k2y(1 − y) + q2x(1 − x) − 2q · kxy −m2 (12)

20

• Anomaly cancellation in the expression for Nµν in Eq. (10) giving

Nµν =1

fπ

e2

2π2S

(

m2

Q2K(m2, Q2)

)

Y µν (13)

• The transition form factor is then

F (q, k) =1

fπ

1

4π2

m2

sK(m2, s) (14)

with

K(m2, s) =

(

ln1 + ρ

1 − ρ− iπ

)2

, ρ =√

1 − 4m2/s, s > 4m2 (15)

For space-like q, with q2 = −Q2 (s = −Q2), with s < 0,

K(m2, Q2) =

(

lnρ+ 1

ρ− 1

)2

, ρ =√

1 + 4m2/Q2 (16)

21

• At large Q2 � m2, F (Q2) is given by:

F (Q2) =1

fπ

1

4π2

m2

Q2

(

lnQ2

m2

)2

(17)

to be compared with the transition form factor for real photon

F (q2 = 0, k2 = 0, p2 = 0) = −(

1

4π2fπ

)

(18)

• As shown below, the behavior of Q2F (Q2) for m = 135MeV fits very

well the CLEO and BaBar data .

22

Q^2*F(Q^2) in GeV

BELLECLEO

BABAR

0.1

0.15

0.2

0.25

0.3

Q^2

*F(Q

^2)(

GeV

)

5 10 15 20 25 30 35 40Q^2(GeV^2)

Figure 11: Chiral anomaly prediction(solid line) for Q2F (Q2) compared with the

BaBar(red) and CLEO(cyan) and the new Belle measured values(blue). The blue

solid line for m = 135MeV and the red curve for m = 120MeV, pQCD prediction

(horizontal line(cyan)) of Brodsly-lepage

23

Figure 12: Dorokhov(2010) similar prediction for Q2F (Q2) (solid curve) compared

with the BaBar and CLEO measured values and the large Q2 pQCD prediction (hor-

izontal dash line) of Brodsky-Lepage

24

3 Conclusion

• Chiral anomaly effects produce, with m = 135MeV the

(m2/Q2)(ln(Q2/m2))2 behavior for the γγ∗ → π0 transition form factor

at Q2 � m2 in contrast with the 2fπ/Q2 behavior given by perturbative

QCD and in striking agreement with the BaBar data at large Q2 and

also with the CLEO data at lower Q2.

• The new Belle results are somewhat below the BaBar values but

qualitatively are not very different from the BaBar data and could be

fitted by lowering the quark mass parameter in the triangle graph from

135MeV to 120MeV.

25

4 Acknowledgments

I would like to thank P. Colangelo, F. De Fazio, the organizers, INFN,

Universities of Lecce and Bari for generous support and warm hospitality

extended to me at Lecce.

26